Place flank steak on a large chopping block; pound steak to 1/4-inch thickness using a meat mallet or rolling pin.
. Combine onion and next 11 ingredients in a bowl; stir in 1/3 cup chicken broth. Spread stuffing over flank steak to within 1/2 inch of edges. Roll up steak, jelly-roll fashion, starting with 1 long side. Secure steak at 2-inch intervals with twine.
. Heat oil in a large nonstick skillet over medium-high heat. Add steak roll; cook 8 minutes, browning on all sides. Add remaining chicken broth and 3 cans tomato sauce.
. Bring to a boil; cover, reduce heat, and simmer 2 hours, turning steak over once. Cut braciola into 4 slices.